Monte-Carlo evaluation of bias and variance in Hurst exponents computed from power spectral analysis of atomic force microscopy topographic images

Abstract

Surface topography influences several surface properties, including friction and adhesion. While a statistical description of surface topography can be obtained from a power spectral density (PSD) analysis of atomic force microscopy (AFM) height maps and fitting the self-affine region of the PSD to determine the Hurst exponent (H), the accuracy of this approach has not been rigorously evaluated yet. Here, we use a Fourier filtering algorithm combined with a novel approach to simulate typical AFM scan-line anisotropy to generate synthetic AFM topography images with known input Hurst exponent. These synthetic AFM images are used as a Monte Carlo experiment to evaluate the variance and bias in H estimation from PSDs across different hypothetical experimental approaches, including the case of a cluster of images collected at one scan size (scale) and the case of a cluster of images collected at different scales. Our analysis reveals that estimates of the Hurst exponent from images collected at a single scale are persistently biased in a scale-dependent fashion despite misleading convergence in variance. This bias can be reduced by combining images collected at least at three different scales across the range of scales accessible to AFM.